Holography, the Second Law and a tilde C-function in higher curvature gravity

We analyze the Second Law of black hole mechanics and the generalization ofthe holographic bound for general theories of gravity. We argue that both thepossibility of defining a holographic bound and the existence of a Second Lawseem to imply each other via the existence of a certain "c-function" (i.e. anever-decreasing function along outgoing null geodesic flow). We are able todefine such a "c-function", that we call \tilde{C}, for general theories ofgravity. It has the nontrivial property of being well defined on generalspacelike surfaces, rather than just on a spatial cross-section of a black holehorizon. We argue that \tilde{C} is a suitable generalization of the notion of"area" in any extension of the holographic bound for general theories ofgravity. Such a function is provided by an algorithm which is similar (althoughnot identical) to that used by Iyer and Wald to define the entropy of adynamical black hole. In a class of higher curvature gravity theories that weanalyze in detail, we are able to prove the monotonicity of \tilde{C} ifseveral physical requirements are satisfied. Apart from the usual ones, theseinclude the cancellation of ghosts in the spectrum of the gravitationalLagrangian. Finally, we point out that our \tilde{C}-function, when evaluatedon a black hole horizon, constitutes by itself an alternative candidate fordefining the entropy of a dynamical black hole.Comment: 28 pages. v4: Published version with minor change